## The area of a circle by slicing

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### Solution

The main idea is to divide a circle into rectangular slices.

\[\begin{gathered}
R^2=x_i^2+y_i^2 \\\\
x_i=\sqrt{R^2-y_i^2}
\end{gathered}\]

The area of the \(i^{\text {th}} \) slice is

\[A_i=2 x_i \Delta y=2 \sqrt{R^2-y_i^2} \Delta y\]

Now let’s calculate the area of the circle:

Let

\[\begin{aligned}
& y=R \sin \theta \\\\
& d y=R \cos \theta d \theta
\end{aligned}\]

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